Particle swarm optimization - Revision history

Fall2024 Wiki Team14: Updated Particle Swarm Optimization (PSO) flowchart.

2024-12-14T23:59:08Z

Updated Particle Swarm Optimization (PSO) flowchart.

← Older revision		Revision as of 19:59, 14 December 2024
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	==Algorithm Discussion==		==Algorithm Discussion==
	The structure of the algorithm is simple. The algorithm seeks to find an optimum of an objective function, <math> f(x) </math>. A set of particles, <math> X </math> with members <math> x \in X </math>, forming a swarm, is initialized randomly. The algorithm runs for a number of iterations. At each iteration, for each particle in the swarm, each particle’s position is updated to be <math> x_{ i+1, p} = x_{i, p} + v_{i, p} </math>. Subscript <math> i </math> indicates the iteration of the algorithm. Subscript <math> p </math> indicates the index of the particle in the swarm. Both the optimal solution for a particular particle, <math> x_{best, p} </math>, and a global optimum, <math> x_{best, g} </math> are tracked. <math> f(x_{best, p}) </math> is the best solution to the objective function that a particular particle, <math> p </math>, has experienced over all iterations. <math> f(x_{best, g}) </math> is the best solution to the objective function for all particles, at all iterations. The algorithm is halted when either a fixed number of iterations is reached or a stopping criteria is reached. A typical stopping criteria is when the change between iterations is below a user-set threshold, demonstrating that the particles are not moving from the identified solution point. The algorithm’s control flow is shown in the following flow chart.		The structure of the algorithm is simple. The algorithm seeks to find an optimum of an objective function, <math> f(x) </math>. A set of particles, <math> X </math> with members <math> x \in X </math>, forming a swarm, is initialized randomly. The algorithm runs for a number of iterations. At each iteration, for each particle in the swarm, each particle’s position is updated to be <math> x_{ i+1, p} = x_{i, p} + v_{i, p} </math>. Subscript <math> i </math> indicates the iteration of the algorithm. Subscript <math> p </math> indicates the index of the particle in the swarm. Both the optimal solution for a particular particle, <math> x_{best, p} </math>, and a global optimum, <math> x_{best, g} </math> are tracked. <math> f(x_{best, p}) </math> is the best solution to the objective function that a particular particle, <math> p </math>, has experienced over all iterations. <math> f(x_{best, g}) </math> is the best solution to the objective function for all particles, at all iterations. The algorithm is halted when either a fixed number of iterations is reached or a stopping criteria is reached. A typical stopping criteria is when the change between iterations is below a user-set threshold, demonstrating that the particles are not moving from the identified solution point. The algorithm’s control flow is shown in the following flow chart.
			[[File:PSO Flow.png\|center\|frameless\|400x400px\|Particle Swarm Optimization Algorithm Flow Chart]]
	[[File:PSO ~~Algorithm~~ Flow.png~~\|400px~~\|center\|frameless\|Particle Swarm Optimization Algorithm Flow Chart]]

	<div style="text-align:center;font-size:1em;">		<div style="text-align:center;font-size:1em;">
	Figure 1: Algorithm Flow Chart		Figure 1: Algorithm Flow Chart

Fall2024 Wiki Team14: Added doi link to reference 17

2024-12-14T17:22:25Z

Added doi link to reference 17

← Older revision		Revision as of 13:22, 14 December 2024
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	<ref> Gudise, V.G. and Venayagamoorthy, G.K. (2003) Comparison of particle swarm optimization and backpropagation as training algorithms for neural networks, ''Proceedings of the 2003 IEEE Swarm Intelligence Symposium'', (pp.110-117) doi: 10.1109/SIS.2003.1202255</ref>.		<ref> Gudise, V.G. and Venayagamoorthy, G.K. (2003) Comparison of particle swarm optimization and backpropagation as training algorithms for neural networks, ''Proceedings of the 2003 IEEE Swarm Intelligence Symposium'', (pp.110-117) doi: 10.1109/SIS.2003.1202255</ref>.
	However, the current state of the current practice for training deep neural networks remains the back-propagation algorithm. Back-propagation is more efficient in the context of neural networks because for each particle at each iteration, there will need to be an error function computation. For example, a swarm size of 25 requires 25 times more error function evaluations		However, the current state of the current practice for training deep neural networks remains the back-propagation algorithm. Back-propagation is more efficient in the context of neural networks because for each particle at each iteration, there will need to be an error function computation. For example, a swarm size of 25 requires 25 times more error function evaluations
	<ref name=wihartiko> Wihartiko, F. (2018) Performance comparison of gentitic algorithms and particle swarm optimization for model integer programming bus timetabling problem, ''IOP Conference Series: Materials Science and Engineering'' doi:10.1088/1757-899X/332/1/012020 </ref>.		<ref name=wihartiko> Wihartiko, F. (2018) Performance comparison of gentitic algorithms and particle swarm optimization for model integer programming bus timetabling problem, ''IOP Conference Series: Materials Science and Engineering'', doi:10.1088/1757-899X/332/1/012020 </ref>.

	In addition to identifying weights in the Neural Networks (training), PSO has been used in selecting hyperparameters for neural networks. In these applications PSO is used to maximize the neural network’s accuracy by varying parameters including number of nodes and structure of the networks		In addition to identifying weights in the Neural Networks (training), PSO has been used in selecting hyperparameters for neural networks. In these applications PSO is used to maximize the neural network’s accuracy by varying parameters including number of nodes and structure of the networks
	<ref> Lorenzo, Pablo et al (2017) Particle swarm optimization for hyperparameter selection in deep neural networks, ~~Association for Computing Machinery~~ </ref>.		<ref> Lorenzo, Pablo et al (2017) Particle swarm optimization for hyperparameter selection in deep neural networks, ''Proceedings of the Genetic and Evolutionary Computation Conference (pp. 481-488) https://doi.org/10.1145/3071178.3071208</ref>.
	PSO fits this application well because this function is nonlinear and the gradient is non-continuous. Hyperparameters for NNs may be real, integer or binary values.		PSO fits this application well because this function is nonlinear and the gradient is non-continuous. Hyperparameters for NNs may be real, integer or binary values.

Fall2024 Wiki Team14: Removed "we" usage in Numerical example section.

2024-12-14T16:04:51Z

Removed "we" usage in Numerical example section.

← Older revision		Revision as of 12:04, 14 December 2024
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	===Step 3: Define Stopping Criteria ===		===Step 3: Define Stopping Criteria ===
	For this example ~~we are choosing~~ a ~~set~~ number of iterations (t):		For this example select a maximum number of iterations (t):

	<div style="text-align:center;font-size:1em;">		<div style="text-align:center;font-size:1em;">
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	Objective Function: <math> f(x,y) = sin(5x+1)+cos(7y-3)+x^2+y^2 \,</math>		Objective Function: <math> f(x,y) = sin(5x+1)+cos(7y-3)+x^2+y^2 \,</math>

	From Step 2, we defined <math> P = 25 \,</math>		From Step 2, the defined number of particles is <math> P = 25 \,</math>


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	For each particle, determine the best value in its own history. This will be called "Personal Best Objective Function" and notated as <math> pbo^i \,</math>. At initialization, there is only one value.		For each particle, determine the best value in its own history. This will be called "Personal Best Objective Function" and notated as <math> pbo^i \,</math>. At initialization, there is only one value.

	In effect ~~we are~~ tracking the position and value of each particle through all iterations and choosing the optimal value - in this example the minimum. As such, rather than tracking all values, the computational memory needs can be reduced by only tracking the optima found so far and comparing to the next iteration’s value:		In effect the program is tracking the position and value of each particle through all iterations and choosing the optimal value - in this example the minimum. As such, rather than tracking all values, the computational memory needs can be reduced by only tracking the optima found so far and comparing to the next iteration’s value:

Fall2024 Wiki Team14: Minor edit

2024-12-14T13:54:17Z

Minor edit

← Older revision		Revision as of 09:54, 14 December 2024
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	==Algorithm Discussion==		==Algorithm Discussion==
	The structure of the algorithm is simple. The algorithm seeks to find an optimum of an objective function, <math> f(x) </math>. A set of particles, <math> X </math> with members <math> x \in X </math>, forming a swarm, is initialized randomly. The algorithm runs for a number of iterations. At each iteration, for each particle in the swarm, each particle’s position is updated to be <math> x_{ i+1, p} = x_{i, p} + v_{i, p} </math>. Subscript <math> i </math> indicates the iteration of the algorithm. Subscript <math> p </math> indicates the index of the particle in the swarm. Both the optimal solution for a particular particle, <math> x_{best, p} </math>, and a global optimum, <math> x_{best, g} </math> are tracked. <math> f(x_{best, p}) </math> is the best solution to the objective function that a particular particle, <math> p </math>, has experienced over all iterations. <math> f(x_{best, g}) </math> is the best solution to the objective function for all particles, at all iterations. The algorithm is halted when either a fixed number of iterations is reached or a stopping criteria is reached. A typical stopping criteria is when the change between iterations is below a user-set threshold, demonstrating that the particles are not moving from the identified solution point. The algorithm’s control flow is shown in the following flow chart.		The structure of the algorithm is simple. The algorithm seeks to find an optimum of an objective function, <math> f(x) </math>. A set of particles, <math> X </math> with members <math> x \in X </math>, forming a swarm, is initialized randomly. The algorithm runs for a number of iterations. At each iteration, for each particle in the swarm, each particle’s position is updated to be <math> x_{ i+1, p} = x_{i, p} + v_{i, p} </math>. Subscript <math> i </math> indicates the iteration of the algorithm. Subscript <math> p </math> indicates the index of the particle in the swarm. Both the optimal solution for a particular particle, <math> x_{best, p} </math>, and a global optimum, <math> x_{best, g} </math> are tracked. <math> f(x_{best, p}) </math> is the best solution to the objective function that a particular particle, <math> p </math>, has experienced over all iterations. <math> f(x_{best, g}) </math> is the best solution to the objective function for all particles, at all iterations. The algorithm is halted when either a fixed number of iterations is reached or a stopping criteria is reached. A typical stopping criteria is when the change between iterations is below a user-set threshold, demonstrating that the particles are not moving from the identified solution point. The algorithm’s control flow is shown in the following flow chart.
	[[File:PSO Flow.png\|center\|frameless~~\|400x400px~~\|Particle Swarm Optimization Algorithm Flow Chart]]
			[[File:PSO Algorithm Flow.png\|400px\|center\|frameless\|Particle Swarm Optimization Algorithm Flow Chart]]

	<div style="text-align:center;font-size:1em;">		<div style="text-align:center;font-size:1em;">
	Figure 1: Algorithm Flow Chart		Figure 1: Algorithm Flow Chart
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	Pymoo is an alternative Python implementation		Pymoo is an alternative Python implementation
	<ref name=pymoo/>.		<ref name=pymoo/>.
	Pyswarms is a compact implementation specifically focused on PSO, whereas Pymoo is a more complete library with multiple ~~objective function~~ implementations. Matlab does have PSO implementation as part of the optimization toolbox		Pyswarms is a compact implementation specifically focused on PSO, whereas Pymoo is a more complete library with multiple optimization algorithm implementations. Matlab does have PSO implementation as part of the optimization toolbox
	<ref> MathWorks (2024) particleswarm, Retrieved 2024 from https://www.mathworks.com/help/gads/particleswarm.html </ref>		<ref> MathWorks (2024) particleswarm, Retrieved 2024 from https://www.mathworks.com/help/gads/particleswarm.html </ref>
	There is a 3rd party Matlab implementation available for free		There is a 3rd party Matlab implementation available for free

Fall2024 Wiki Team14: Updated PSO flow diagram

2024-12-14T13:52:15Z

Updated PSO flow diagram

← Older revision		Revision as of 09:52, 14 December 2024
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	==Algorithm Discussion==		==Algorithm Discussion==
	The structure of the algorithm is simple. The algorithm seeks to find an optimum of an objective function, <math> f(x) </math>. A set of particles, <math> X </math> with members <math> x \in X </math>, forming a swarm, is initialized randomly. The algorithm runs for a number of iterations. At each iteration, for each particle in the swarm, each particle’s position is updated to be <math> x_{ i+1, p} = x_{i, p} + v_{i, p} </math>. Subscript <math> i </math> indicates the iteration of the algorithm. Subscript <math> p </math> indicates the index of the particle in the swarm. Both the optimal solution for a particular particle, <math> x_{best, p} </math>, and a global optimum, <math> x_{best, g} </math> are tracked. <math> f(x_{best, p}) </math> is the best solution to the objective function that a particular particle, <math> p </math>, has experienced over all iterations. <math> f(x_{best, g}) </math> is the best solution to the objective function for all particles, at all iterations. The algorithm is halted when either a fixed number of iterations is reached or a stopping criteria is reached. A typical stopping criteria is when the change between iterations is below a user-set threshold, demonstrating that the particles are not moving from the identified solution point. The algorithm’s control flow is shown in the following flow chart.		The structure of the algorithm is simple. The algorithm seeks to find an optimum of an objective function, <math> f(x) </math>. A set of particles, <math> X </math> with members <math> x \in X </math>, forming a swarm, is initialized randomly. The algorithm runs for a number of iterations. At each iteration, for each particle in the swarm, each particle’s position is updated to be <math> x_{ i+1, p} = x_{i, p} + v_{i, p} </math>. Subscript <math> i </math> indicates the iteration of the algorithm. Subscript <math> p </math> indicates the index of the particle in the swarm. Both the optimal solution for a particular particle, <math> x_{best, p} </math>, and a global optimum, <math> x_{best, g} </math> are tracked. <math> f(x_{best, p}) </math> is the best solution to the objective function that a particular particle, <math> p </math>, has experienced over all iterations. <math> f(x_{best, g}) </math> is the best solution to the objective function for all particles, at all iterations. The algorithm is halted when either a fixed number of iterations is reached or a stopping criteria is reached. A typical stopping criteria is when the change between iterations is below a user-set threshold, demonstrating that the particles are not moving from the identified solution point. The algorithm’s control flow is shown in the following flow chart.
			[[File:PSO Flow.png\|center\|frameless\|400x400px\|Particle Swarm Optimization Algorithm Flow Chart]]
	[[File:PSO ~~Algorithm~~ Flow.png~~\|400px~~\|center\|frameless\|Particle Swarm Optimization Algorithm Flow Chart]]

	<div style="text-align:center;font-size:1em;">		<div style="text-align:center;font-size:1em;">
	Figure 1: Algorithm Flow Chart		Figure 1: Algorithm Flow Chart
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	However, the algorithm is simple so many prefer to implement the algorithm directly.		However, the algorithm is simple so many prefer to implement the algorithm directly.

	Particle swarm techniques have seen wide spread application in a variety of industries, notably in the subfield of temperature prediction and control. Huihui Yu et al. investigate the challenge of temperature prediction in Chinese solar greenhouses, which are critical for year-round crop production in harsh winter conditions. The study explores various computational models, concluding that a modified PSO approach outperforms traditional methods like physical modeling, time-series models, neural networks, and support vector machines as they can not handle the complex, nonlinear, and dynamic greenhouse environment. Performance was validated using environmental data collected from 500 greenhouses in the Shandong Province, China and ultimately concludes that the proposed modified PSO model is an effective tool for solar greenhouse temperature prediction <ref>uihui Yu, Yingyi Chen, Shahbaz Gul Hassan, Daoliang Li,Prediction of the temperature in a Chinese solar greenhouse based on LSSVM optimized by improved PSO,Computers and Electronics in Agriculture,Volume 122,2016,Pages 94-102,ISSN 0168-1699,

	<nowiki>https://doi.org/10.1016/j.compag.2016.01.019</nowiki> (<nowiki>https://www.sciencedirect.com/science/article/pii/S0168169916000247</nowiki>)</ref>. Similarly, Majid Sivashi et al. found PSO an effective model to help with oil recovery. PSO was applied to optimize water injection temperature, which plays a crucial role in enhancing oil recovery by reducing oil viscosity and improving its mobility. Unlike conventional assumptions that the highest injection temperature is always optimal, PSO found nuanced temperature settings tailored to the specific conditions of the reservoir, balancing oil recovery and energy up to 7% more efficiently <ref>Majid Siavashi, Mohammad Hossein Doranehgard,Particle swarm optimization of thermal enhanced oil recovery from oilfields with temperature control, Applied Thermal Engineering,Volume 123,2017,Pages 658-669,ISSN 1359-4311,<nowiki>https://doi.org/10.1016/j.applthermaleng.2017.05.109</nowiki>.</ref>.

	PSO has also seen some application in the medical field. This research focuses on improving the automated diagnosis of Acute Lymphoblastic Leukaemia through intelligent decision support systems using evolutionary feature optimization techniques. The study proposes two modified Particle Swarm Optimization algorithms to enhance the identification of healthy and abnormal lymphocytic cells. The modified PSO algorithms incorporate innovative search mechanisms inspired by biological behaviors, such as attraction to a food source and avoidance of predators, to enhance the diversification of the search process and escape premature convergence. The algorithms were evaluated using the ALL-IDB2 database, achieving high classification accuracy with average GM scores of 94.94% and 96.25% for the first and second algorithms, respectively and also demonstrated faster convergence rates than other common search models such as a genetic algorithm <ref>Worawut Srisukkham, Li Zhang, Siew Chin Neoh, Stephen Todryk, Chee Peng Lim,Intelligent leukaemia diagnosis with bare-bones PSO based feature optimization,Applied Soft Computing Volume 56,2017,Pages 405-419,ISSN 1568-4946,<nowiki>https://doi.org/10.1016/j.asoc.2017.03.024.(https://www.sciencedirect.com/science/article/pii/S1568494617301485)</nowiki></ref>.

	A common theme in studies that apply PSO to real world problems is that the algorithm is often modified and integrated with other models to enhance its performance and adapt it to the specific challenges of a given problem. This is due to some of the limitations of PSO, a significant issue of premature convergence, where particles settle in suboptimal solutions, particularly in highly nonlinear, multimodal, or high-dimensional search spaces. To counteract this, PSO is often combined with mechanisms like chaotic maps, mutation operators, or genetic algorithms to maintain diversity in the swarm. PSO is also often combined with ML algorithms, namely SVM and Neural Networks, by optimizing hyper-parameters or feature selection for classification tasks. An advantage of PSO lies in its ability to be simply combined with other techniques. This stems from PSO's small parameter set, advantage in global exploration, and ability to parallelize computations which makes it ideal for integration with more computationally intensive models, ensuring scalability for large and complex datasets.

	==Conclusions==		==Conclusions==

Fall2024 Wiki Team14 at 07:02, 14 December 2024

2024-12-14T07:02:55Z

← Older revision		Revision as of 03:02, 14 December 2024
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	to leading to a technique dependent only on simple calculations and a number of starting points.		to leading to a technique dependent only on simple calculations and a number of starting points.

	The PSO algorithm was formulated and initially presented by Kennedy and Eberhart specifically to find solutions to discontinuous, non-differentiable functions. The concept was presented at the IEEE conference on neural networks in 1995		The PSO algorithm was formulated and initially presented by Kennedy and Eberhart specifically to find solutions to discontinuous, but non-differentiable functions. The concept was presented at the IEEE conference on neural networks in 1995
	<ref name=Kennedy1> Kennedy, J. and Eberhart, R. (1995) Particle Swarm Optimization, ''Proceedings of the IEEE International Conference on Neural Networks'', (pp. 1942-1945), doi: 10.1109/ICNN.1995.488968 </ref>.		<ref name=Kennedy1> Kennedy, J. and Eberhart, R. (1995) Particle Swarm Optimization, ''Proceedings of the IEEE International Conference on Neural Networks'', (pp. 1942-1945), doi: 10.1109/ICNN.1995.488968 </ref>.
	The pair later published a book Swarm Intelligence in 2001 that further expanded on the concept		The pair later published a book Swarm Intelligence in 2001 that further expanded on the concept
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	and electric power distribution		and electric power distribution
	<ref> Sharmeela, C. et al (2022) IoT, Machine Learning and Blockchain Technologies for Renewable Energy and Modern Hybrid Power Systems, River Publishers </ref>.		<ref> Sharmeela, C. et al (2022) IoT, Machine Learning and Blockchain Technologies for Renewable Energy and Modern Hybrid Power Systems, River Publishers </ref>.
	As functions get complex in the real-world, the PSO algorithm is still capable of performing an effective search within very complex functions due to its heuristic and operability on non-differentiable functions.		As functions get very complex in the real-world, the PSO algorithm is still capable of performing an effective search within very complex functions due to its heuristic and operability on non-differentiable functions.

	==Algorithm Discussion==		==Algorithm Discussion==
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	However, the algorithm is simple so many prefer to implement the algorithm directly.		However, the algorithm is simple so many prefer to implement the algorithm directly.

			Particle swarm techniques have seen wide spread application in a variety of industries, notably in the subfield of temperature prediction and control. Huihui Yu et al. investigate the challenge of temperature prediction in Chinese solar greenhouses, which are critical for year-round crop production in harsh winter conditions. The study explores various computational models, concluding that a modified PSO approach outperforms traditional methods like physical modeling, time-series models, neural networks, and support vector machines as they can not handle the complex, nonlinear, and dynamic greenhouse environment. Performance was validated using environmental data collected from 500 greenhouses in the Shandong Province, China and ultimately concludes that the proposed modified PSO model is an effective tool for solar greenhouse temperature prediction <ref>uihui Yu, Yingyi Chen, Shahbaz Gul Hassan, Daoliang Li,Prediction of the temperature in a Chinese solar greenhouse based on LSSVM optimized by improved PSO,Computers and Electronics in Agriculture,Volume 122,2016,Pages 94-102,ISSN 0168-1699,

			<nowiki>https://doi.org/10.1016/j.compag.2016.01.019</nowiki> (<nowiki>https://www.sciencedirect.com/science/article/pii/S0168169916000247</nowiki>)</ref>. Similarly, Majid Sivashi et al. found PSO an effective model to help with oil recovery. PSO was applied to optimize water injection temperature, which plays a crucial role in enhancing oil recovery by reducing oil viscosity and improving its mobility. Unlike conventional assumptions that the highest injection temperature is always optimal, PSO found nuanced temperature settings tailored to the specific conditions of the reservoir, balancing oil recovery and energy up to 7% more efficiently <ref>Majid Siavashi, Mohammad Hossein Doranehgard,Particle swarm optimization of thermal enhanced oil recovery from oilfields with temperature control, Applied Thermal Engineering,Volume 123,2017,Pages 658-669,ISSN 1359-4311,<nowiki>https://doi.org/10.1016/j.applthermaleng.2017.05.109</nowiki>.</ref>.

			PSO has also seen some application in the medical field. This research focuses on improving the automated diagnosis of Acute Lymphoblastic Leukaemia through intelligent decision support systems using evolutionary feature optimization techniques. The study proposes two modified Particle Swarm Optimization algorithms to enhance the identification of healthy and abnormal lymphocytic cells. The modified PSO algorithms incorporate innovative search mechanisms inspired by biological behaviors, such as attraction to a food source and avoidance of predators, to enhance the diversification of the search process and escape premature convergence. The algorithms were evaluated using the ALL-IDB2 database, achieving high classification accuracy with average GM scores of 94.94% and 96.25% for the first and second algorithms, respectively and also demonstrated faster convergence rates than other common search models such as a genetic algorithm <ref>Worawut Srisukkham, Li Zhang, Siew Chin Neoh, Stephen Todryk, Chee Peng Lim,Intelligent leukaemia diagnosis with bare-bones PSO based feature optimization,Applied Soft Computing Volume 56,2017,Pages 405-419,ISSN 1568-4946,<nowiki>https://doi.org/10.1016/j.asoc.2017.03.024.(https://www.sciencedirect.com/science/article/pii/S1568494617301485)</nowiki></ref>.

			A common theme in studies that apply PSO to real world problems is that the algorithm is often modified and integrated with other models to enhance its performance and adapt it to the specific challenges of a given problem. This is due to some of the limitations of PSO, a significant issue of premature convergence, where particles settle in suboptimal solutions, particularly in highly nonlinear, multimodal, or high-dimensional search spaces. To counteract this, PSO is often combined with mechanisms like chaotic maps, mutation operators, or genetic algorithms to maintain diversity in the swarm. PSO is also often combined with ML algorithms, namely SVM and Neural Networks, by optimizing hyper-parameters or feature selection for classification tasks. An advantage of PSO lies in its ability to be simply combined with other techniques. This stems from PSO's small parameter set, advantage in global exploration, and ability to parallelize computations which makes it ideal for integration with more computationally intensive models, ensuring scalability for large and complex datasets.

	==Conclusions==		==Conclusions==

Fall2024 Wiki Team14: Minior grammar

2024-12-14T04:57:19Z

Minior grammar

Fall2024 Wiki Team14: /* Step 3: Define Stopping Criteria */

2024-12-14T03:54:29Z

Step 3: Define Stopping Criteria

← Older revision		Revision as of 23:54, 13 December 2024
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	===Step 3: Define Stopping Criteria ===		===Step 3: Define Stopping Criteria ===
	For this example ~~we are choosing~~ a set number of iterations (t):		For this example a set number of iterations (t) is chosen:

	<div style="text-align:center;font-size:1em;">		<div style="text-align:center;font-size:1em;">

Fall2024 Wiki Team14: /* Algorithm Discussion */

2024-12-13T22:00:02Z

Algorithm Discussion

← Older revision		Revision as of 18:00, 13 December 2024
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	<math> v_{particle} </math> initialized to 0		<math> v_{particle} </math> initialized to 0

	<math> x_{best, p} </math> initialized		<math> x_{best, p} </math> initialized to <math> \infty </math>

	<math> x_{best, g} </math> initialized to		<math> x_{best, g} </math> initialized to <math> \infty </math>

	For i in MAX_ITERATIONS:		For i in MAX_ITERATIONS:

Fall2024 Wiki Team14: /* Algorithm Discussion */

2024-12-13T21:49:29Z

Algorithm Discussion

← Older revision		Revision as of 17:49, 13 December 2024
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	<div style="text-align:center;font-size:1em;">		<div style="text-align:center;font-size:1em;">
	Figure 2: Algorithm Pseudocode		Figure 2: Algorithm Pseudocode
	<div>		</div>

	The algorithm has a few hyperparameters including; number of particles in the swarm, weights of the cognitive update on the velocity, weights on the social update on the velocity. The Pymoo Python implementation suggests a swarm size of 25, velocity weight of 0.8, a social update weight of 2.0 and a cognitive update of 2.0		The algorithm has a few hyperparameters including; number of particles in the swarm, weights of the cognitive update on the velocity, weights on the social update on the velocity. The Pymoo Python implementation suggests a swarm size of 25, velocity weight of 0.8, a social update weight of 2.0 and a cognitive update of 2.0

← Older revision		Revision as of 00:57, 14 December 2024
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	to leading to a technique dependent only on simple calculations and a number of starting points.		to leading to a technique dependent only on simple calculations and a number of starting points.

	The PSO algorithm was formulated and initially presented by Kennedy and Eberhart specifically to find solutions to discontinuous, ~~but~~ non-differentiable functions. The concept was presented at the IEEE conference on neural networks in 1995		The PSO algorithm was formulated and initially presented by Kennedy and Eberhart specifically to find solutions to discontinuous, non-differentiable functions. The concept was presented at the IEEE conference on neural networks in 1995
	<ref name=Kennedy1> Kennedy, J. and Eberhart, R. (1995) Particle Swarm Optimization, ''Proceedings of the IEEE International Conference on Neural Networks'', (pp. 1942-1945), doi: 10.1109/ICNN.1995.488968 </ref>.		<ref name=Kennedy1> Kennedy, J. and Eberhart, R. (1995) Particle Swarm Optimization, ''Proceedings of the IEEE International Conference on Neural Networks'', (pp. 1942-1945), doi: 10.1109/ICNN.1995.488968 </ref>.
	The pair later published a book Swarm Intelligence in 2001 that further expanded on the concept		The pair later published a book Swarm Intelligence in 2001 that further expanded on the concept
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	and electric power distribution		and electric power distribution
	<ref> Sharmeela, C. et al (2022) IoT, Machine Learning and Blockchain Technologies for Renewable Energy and Modern Hybrid Power Systems, River Publishers </ref>.		<ref> Sharmeela, C. et al (2022) IoT, Machine Learning and Blockchain Technologies for Renewable Energy and Modern Hybrid Power Systems, River Publishers </ref>.
	As functions get ~~very~~ complex in the real-world, the PSO algorithm is still capable of performing an effective search within very complex functions due to its heuristic and operability on non-differentiable functions.		As functions get complex in the real-world, the PSO algorithm is still capable of performing an effective search within very complex functions due to its heuristic and operability on non-differentiable functions.

	==Algorithm Discussion==		==Algorithm Discussion==
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	===Step 3: Define Stopping Criteria ===		===Step 3: Define Stopping Criteria ===
	For this example a set number of iterations (t) ~~is chosen~~:		For this example we are choosing a set number of iterations (t):

	<div style="text-align:center;font-size:1em;">		<div style="text-align:center;font-size:1em;">